Integrand size = 25, antiderivative size = 216 \[ \int \frac {1}{d+e x^2+\frac {a+b x^2+c x^4}{x^2}} \, dx=-\frac {\sqrt {b+d-\sqrt {b^2+2 b d+d^2-4 a (c+e)}} \arctan \left (\frac {\sqrt {2} \sqrt {c+e} x}{\sqrt {b+d-\sqrt {b^2+2 b d+d^2-4 a (c+e)}}}\right )}{\sqrt {2} \sqrt {c+e} \sqrt {b^2+2 b d+d^2-4 a (c+e)}}+\frac {\sqrt {b+d+\sqrt {b^2+2 b d+d^2-4 a (c+e)}} \arctan \left (\frac {\sqrt {2} \sqrt {c+e} x}{\sqrt {b+d+\sqrt {b^2+2 b d+d^2-4 a (c+e)}}}\right )}{\sqrt {2} \sqrt {c+e} \sqrt {b^2+2 b d+d^2-4 a (c+e)}} \] Output:
-1/2*(b+d-(b^2+2*b*d+d^2-4*a*(c+e))^(1/2))^(1/2)*arctan(2^(1/2)*(c+e)^(1/2 )*x/(b+d-(b^2+2*b*d+d^2-4*a*(c+e))^(1/2))^(1/2))*2^(1/2)/(c+e)^(1/2)/(b^2+ 2*b*d+d^2-4*a*(c+e))^(1/2)+1/2*(b+d+(b^2+2*b*d+d^2-4*a*(c+e))^(1/2))^(1/2) *arctan(2^(1/2)*(c+e)^(1/2)*x/(b+d+(b^2+2*b*d+d^2-4*a*(c+e))^(1/2))^(1/2)) *2^(1/2)/(c+e)^(1/2)/(b^2+2*b*d+d^2-4*a*(c+e))^(1/2)
Time = 0.06 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.12 \[ \int \frac {1}{d+e x^2+\frac {a+b x^2+c x^4}{x^2}} \, dx=\frac {\left (-b-d+\sqrt {b^2+2 b d+d^2-4 a (c+e)}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c+e} x}{\sqrt {b+d-\sqrt {b^2+2 b d+d^2-4 a (c+e)}}}\right )+\sqrt {b+d-\sqrt {b^2+2 b d+d^2-4 a (c+e)}} \sqrt {b+d+\sqrt {b^2+2 b d+d^2-4 a (c+e)}} \arctan \left (\frac {\sqrt {2} \sqrt {c+e} x}{\sqrt {b+d+\sqrt {b^2+2 b d+d^2-4 a (c+e)}}}\right )}{\sqrt {2} \sqrt {c+e} \sqrt {b^2+2 b d+d^2-4 a (c+e)} \sqrt {b+d-\sqrt {b^2+2 b d+d^2-4 a (c+e)}}} \] Input:
Integrate[(d + e*x^2 + (a + b*x^2 + c*x^4)/x^2)^(-1),x]
Output:
((-b - d + Sqrt[b^2 + 2*b*d + d^2 - 4*a*(c + e)])*ArcTan[(Sqrt[2]*Sqrt[c + e]*x)/Sqrt[b + d - Sqrt[b^2 + 2*b*d + d^2 - 4*a*(c + e)]]] + Sqrt[b + d - Sqrt[b^2 + 2*b*d + d^2 - 4*a*(c + e)]]*Sqrt[b + d + Sqrt[b^2 + 2*b*d + d^ 2 - 4*a*(c + e)]]*ArcTan[(Sqrt[2]*Sqrt[c + e]*x)/Sqrt[b + d + Sqrt[b^2 + 2 *b*d + d^2 - 4*a*(c + e)]]])/(Sqrt[2]*Sqrt[c + e]*Sqrt[b^2 + 2*b*d + d^2 - 4*a*(c + e)]*Sqrt[b + d - Sqrt[b^2 + 2*b*d + d^2 - 4*a*(c + e)]])
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\frac {a+b x^2+c x^4}{x^2}+d+e x^2} \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \frac {1}{\frac {a+b x^2+c x^4}{x^2}+d+e x^2}dx\) |
Input:
Int[(d + e*x^2 + (a + b*x^2 + c*x^4)/x^2)^(-1),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.25
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (c +e \right ) \textit {\_Z}^{4}+\left (b +d \right ) \textit {\_Z}^{2}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +2 \textit {\_R}^{3} e +\textit {\_R} b +\textit {\_R} d}\right )}{2}\) | \(54\) |
| default | \(\left (4 c +4 e \right ) \left (\frac {\left (\sqrt {-4 a c -4 a e +b^{2}+2 b d +d^{2}}-b -d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-2 c -2 e \right ) x \sqrt {2}}{2 \sqrt {\left (\sqrt {-4 a c -4 a e +b^{2}+2 b d +d^{2}}-b -d \right ) \left (c +e \right )}}\right )}{8 \sqrt {-4 a c -4 a e +b^{2}+2 b d +d^{2}}\, \left (c +e \right ) \sqrt {\left (\sqrt {-4 a c -4 a e +b^{2}+2 b d +d^{2}}-b -d \right ) \left (c +e \right )}}+\frac {\left (\sqrt {-4 a c -4 a e +b^{2}+2 b d +d^{2}}+b +d \right ) \sqrt {2}\, \arctan \left (\frac {\left (2 c +2 e \right ) x \sqrt {2}}{2 \sqrt {\left (c +e \right ) \left (\sqrt {-4 a c -4 a e +b^{2}+2 b d +d^{2}}+b +d \right )}}\right )}{8 \sqrt {-4 a c -4 a e +b^{2}+2 b d +d^{2}}\, \left (c +e \right ) \sqrt {\left (c +e \right ) \left (\sqrt {-4 a c -4 a e +b^{2}+2 b d +d^{2}}+b +d \right )}}\right )\) | \(280\) |
Input:
int(1/(d+e*x^2+(c*x^4+b*x^2+a)/x^2),x,method=_RETURNVERBOSE)
Output:
1/2*sum(_R^2/(2*_R^3*c+2*_R^3*e+_R*b+_R*d)*ln(x-_R),_R=RootOf((c+e)*_Z^4+( b+d)*_Z^2+a))
Leaf count of result is larger than twice the leaf count of optimal. 1995 vs. \(2 (181) = 362\).
Time = 0.09 (sec) , antiderivative size = 1995, normalized size of antiderivative = 9.24 \[ \int \frac {1}{d+e x^2+\frac {a+b x^2+c x^4}{x^2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(d+e*x^2+(c*x^4+b*x^2+a)/x^2),x, algorithm="fricas")
Output:
1/2*sqrt(1/2)*sqrt(-(b + d + (b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2 - 4*a*e^2 + (b^2 - 8*a*c + 2*b*d + d^2)*e)/sqrt(b^2*c^2 - 4*a*c^3 + 2*b*c^2*d + c^2* d^2 - 4*a*e^3 + (b^2 - 12*a*c + 2*b*d + d^2)*e^2 + 2*(b^2*c - 6*a*c^2 + 2* b*c*d + c*d^2)*e))/(b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2 - 4*a*e^2 + (b^2 - 8 *a*c + 2*b*d + d^2)*e))*log(sqrt(1/2)*(b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2 - 4*a*e^2 + (b^2 - 8*a*c + 2*b*d + d^2)*e)*sqrt(-(b + d + (b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2 - 4*a*e^2 + (b^2 - 8*a*c + 2*b*d + d^2)*e)/sqrt(b^2*c^2 - 4*a*c^3 + 2*b*c^2*d + c^2*d^2 - 4*a*e^3 + (b^2 - 12*a*c + 2*b*d + d^2)*e ^2 + 2*(b^2*c - 6*a*c^2 + 2*b*c*d + c*d^2)*e))/(b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2 - 4*a*e^2 + (b^2 - 8*a*c + 2*b*d + d^2)*e))/sqrt(b^2*c^2 - 4*a*c^3 + 2*b*c^2*d + c^2*d^2 - 4*a*e^3 + (b^2 - 12*a*c + 2*b*d + d^2)*e^2 + 2*(b ^2*c - 6*a*c^2 + 2*b*c*d + c*d^2)*e) + x) - 1/2*sqrt(1/2)*sqrt(-(b + d + ( b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2 - 4*a*e^2 + (b^2 - 8*a*c + 2*b*d + d^2)* e)/sqrt(b^2*c^2 - 4*a*c^3 + 2*b*c^2*d + c^2*d^2 - 4*a*e^3 + (b^2 - 12*a*c + 2*b*d + d^2)*e^2 + 2*(b^2*c - 6*a*c^2 + 2*b*c*d + c*d^2)*e))/(b^2*c - 4* a*c^2 + 2*b*c*d + c*d^2 - 4*a*e^2 + (b^2 - 8*a*c + 2*b*d + d^2)*e))*log(-s qrt(1/2)*(b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2 - 4*a*e^2 + (b^2 - 8*a*c + 2*b *d + d^2)*e)*sqrt(-(b + d + (b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2 - 4*a*e^2 + (b^2 - 8*a*c + 2*b*d + d^2)*e)/sqrt(b^2*c^2 - 4*a*c^3 + 2*b*c^2*d + c^2*d ^2 - 4*a*e^3 + (b^2 - 12*a*c + 2*b*d + d^2)*e^2 + 2*(b^2*c - 6*a*c^2 + ...
Time = 7.39 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.74 \[ \int \frac {1}{d+e x^2+\frac {a+b x^2+c x^4}{x^2}} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{2} c^{3} + 768 a^{2} c^{2} e + 768 a^{2} c e^{2} + 256 a^{2} e^{3} - 128 a b^{2} c^{2} - 256 a b^{2} c e - 128 a b^{2} e^{2} - 256 a b c^{2} d - 512 a b c d e - 256 a b d e^{2} - 128 a c^{2} d^{2} - 256 a c d^{2} e - 128 a d^{2} e^{2} + 16 b^{4} c + 16 b^{4} e + 64 b^{3} c d + 64 b^{3} d e + 96 b^{2} c d^{2} + 96 b^{2} d^{2} e + 64 b c d^{3} + 64 b d^{3} e + 16 c d^{4} + 16 d^{4} e\right ) + t^{2} \left (- 16 a b c - 16 a b e - 16 a c d - 16 a d e + 4 b^{3} + 12 b^{2} d + 12 b d^{2} + 4 d^{3}\right ) + a, \left ( t \mapsto t \log {\left (64 t^{3} a c^{2} + 128 t^{3} a c e + 64 t^{3} a e^{2} - 16 t^{3} b^{2} c - 16 t^{3} b^{2} e - 32 t^{3} b c d - 32 t^{3} b d e - 16 t^{3} c d^{2} - 16 t^{3} d^{2} e - 2 t b - 2 t d + x \right )} \right )\right )} \] Input:
integrate(1/(d+e*x**2+(c*x**4+b*x**2+a)/x**2),x)
Output:
RootSum(_t**4*(256*a**2*c**3 + 768*a**2*c**2*e + 768*a**2*c*e**2 + 256*a** 2*e**3 - 128*a*b**2*c**2 - 256*a*b**2*c*e - 128*a*b**2*e**2 - 256*a*b*c**2 *d - 512*a*b*c*d*e - 256*a*b*d*e**2 - 128*a*c**2*d**2 - 256*a*c*d**2*e - 1 28*a*d**2*e**2 + 16*b**4*c + 16*b**4*e + 64*b**3*c*d + 64*b**3*d*e + 96*b* *2*c*d**2 + 96*b**2*d**2*e + 64*b*c*d**3 + 64*b*d**3*e + 16*c*d**4 + 16*d* *4*e) + _t**2*(-16*a*b*c - 16*a*b*e - 16*a*c*d - 16*a*d*e + 4*b**3 + 12*b* *2*d + 12*b*d**2 + 4*d**3) + a, Lambda(_t, _t*log(64*_t**3*a*c**2 + 128*_t **3*a*c*e + 64*_t**3*a*e**2 - 16*_t**3*b**2*c - 16*_t**3*b**2*e - 32*_t**3 *b*c*d - 32*_t**3*b*d*e - 16*_t**3*c*d**2 - 16*_t**3*d**2*e - 2*_t*b - 2*_ t*d + x)))
\[ \int \frac {1}{d+e x^2+\frac {a+b x^2+c x^4}{x^2}} \, dx=\int { \frac {1}{e x^{2} + d + \frac {c x^{4} + b x^{2} + a}{x^{2}}} \,d x } \] Input:
integrate(1/(d+e*x^2+(c*x^4+b*x^2+a)/x^2),x, algorithm="maxima")
Output:
integrate(1/(e*x^2 + d + (c*x^4 + b*x^2 + a)/x^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 3423 vs. \(2 (181) = 362\).
Time = 20.09 (sec) , antiderivative size = 3423, normalized size of antiderivative = 15.85 \[ \int \frac {1}{d+e x^2+\frac {a+b x^2+c x^4}{x^2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(d+e*x^2+(c*x^4+b*x^2+a)/x^2),x, algorithm="giac")
Output:
-1/4*(2*b^2*c^2 - 8*a*c^3 + 4*b*c^2*d + 2*c^2*d^2 + 4*b^2*c*e - 24*a*c^2*e + 8*b*c*d*e + 4*c*d^2*e + 2*b^2*e^2 - 24*a*c*e^2 + 4*b*d*e^2 + 2*d^2*e^2 - 8*a*e^3 - sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e)*sqrt(b*c + c*d + b*e + d*e + sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e)*(c + e))*b^2 + 4*sq rt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e)*sqrt(b*c + c*d + b*e + d*e + sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e)*(c + e))*a*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e)*sqrt(b*c + c*d + b*e + d*e + sqrt(b^2 - 4* a*c + 2*b*d + d^2 - 4*a*e)*(c + e))*b*c - sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e)*sqrt(b*c + c*d + b*e + d*e + sqrt(b^2 - 4*a*c + 2*b*d + d^ 2 - 4*a*e)*(c + e))*c^2 - 2*sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e )*sqrt(b*c + c*d + b*e + d*e + sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e)*(c + e))*b*d + 2*sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e)*sqrt(b*c + c *d + b*e + d*e + sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e)*(c + e))*c*d - sq rt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e)*sqrt(b*c + c*d + b*e + d*e + sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e)*(c + e))*d^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e)*sqrt(b*c + c*d + b*e + d*e + sqrt(b^2 - 4* a*c + 2*b*d + d^2 - 4*a*e)*(c + e))*a*e + 2*sqrt(2)*sqrt(b^2 - 4*a*c + 2*b *d + d^2 - 4*a*e)*sqrt(b*c + c*d + b*e + d*e + sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e)*(c + e))*b*e - 2*sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a *e)*sqrt(b*c + c*d + b*e + d*e + sqrt(b^2 - 4*a*c + 2*b*d + d^2 - 4*a*e...
Time = 0.93 (sec) , antiderivative size = 1956, normalized size of antiderivative = 9.06 \[ \int \frac {1}{d+e x^2+\frac {a+b x^2+c x^4}{x^2}} \, dx=\text {Too large to display} \] Input:
int(1/(d + (a + b*x^2 + c*x^4)/x^2 + e*x^2),x)
Output:
2*atanh((2*(x*(2*b^2*c - 4*a*c^2 - 4*a*e^2 + 2*c*d^2 + 2*b^2*e + 2*d^2*e - 8*a*c*e + 4*b*c*d + 4*b*d*e) - (x*(((2*b*d - 4*a*e - 4*a*c + b^2 + d^2)^3 )^(1/2) + 3*b*d^2 + 3*b^2*d + b^3 + d^3 - 4*a*b*c - 4*a*b*e - 4*a*c*d - 4* a*d*e)*(8*b^3*c^2 + 8*b^3*e^2 + 8*c^2*d^3 + 8*d^3*e^2 + 24*b*c^2*d^2 + 24* b^2*c^2*d + 24*b*d^2*e^2 + 24*b^2*d*e^2 - 32*a*b*c^3 - 32*a*b*e^3 - 32*a*c ^3*d - 32*a*d*e^3 + 16*b^3*c*e + 16*c*d^3*e - 96*a*b*c*e^2 - 96*a*b*c^2*e - 96*a*c*d*e^2 - 96*a*c^2*d*e + 48*b*c*d^2*e + 48*b^2*c*d*e))/(8*(b^4*c + c*d^4 + b^4*e + d^4*e + 16*a^2*c^3 + 16*a^2*e^3 - 8*a*b^2*c^2 - 8*a*b^2*e^ 2 - 8*a*c^2*d^2 + 48*a^2*c*e^2 + 48*a^2*c^2*e + 6*b^2*c*d^2 - 8*a*d^2*e^2 + 6*b^2*d^2*e + 4*b*c*d^3 + 4*b^3*c*d + 4*b*d^3*e + 4*b^3*d*e - 16*a*b*c^2 *d - 16*a*b^2*c*e - 16*a*b*d*e^2 - 16*a*c*d^2*e - 32*a*b*c*d*e)))*(-(((2*b *d - 4*a*e - 4*a*c + b^2 + d^2)^3)^(1/2) + 3*b*d^2 + 3*b^2*d + b^3 + d^3 - 4*a*b*c - 4*a*b*e - 4*a*c*d - 4*a*d*e)/(8*(b^4*c + c*d^4 + b^4*e + d^4*e + 16*a^2*c^3 + 16*a^2*e^3 - 8*a*b^2*c^2 - 8*a*b^2*e^2 - 8*a*c^2*d^2 + 48*a ^2*c*e^2 + 48*a^2*c^2*e + 6*b^2*c*d^2 - 8*a*d^2*e^2 + 6*b^2*d^2*e + 4*b*c* d^3 + 4*b^3*c*d + 4*b*d^3*e + 4*b^3*d*e - 16*a*b*c^2*d - 16*a*b^2*c*e - 16 *a*b*d*e^2 - 16*a*c*d^2*e - 32*a*b*c*d*e)))^(1/2))/(2*a*c + 2*a*e))*(-(((2 *b*d - 4*a*e - 4*a*c + b^2 + d^2)^3)^(1/2) + 3*b*d^2 + 3*b^2*d + b^3 + d^3 - 4*a*b*c - 4*a*b*e - 4*a*c*d - 4*a*d*e)/(8*(b^4*c + c*d^4 + b^4*e + d^4* e + 16*a^2*c^3 + 16*a^2*e^3 - 8*a*b^2*c^2 - 8*a*b^2*e^2 - 8*a*c^2*d^2 +...
Time = 0.66 (sec) , antiderivative size = 938, normalized size of antiderivative = 4.34 \[ \int \frac {1}{d+e x^2+\frac {a+b x^2+c x^4}{x^2}} \, dx =\text {Too large to display} \] Input:
int(1/(d+e*x^2+(c*x^4+b*x^2+a)/x^2),x)
Output:
(2*sqrt(c + e)*sqrt(2*sqrt(a)*sqrt(c + e) + b + d)*atan((sqrt(2*sqrt(a)*sq rt(c + e) - b - d) - 2*sqrt(c + e)*x)/sqrt(2*sqrt(a)*sqrt(c + e) + b + d)) *b + 2*sqrt(c + e)*sqrt(2*sqrt(a)*sqrt(c + e) + b + d)*atan((sqrt(2*sqrt(a )*sqrt(c + e) - b - d) - 2*sqrt(c + e)*x)/sqrt(2*sqrt(a)*sqrt(c + e) + b + d))*d - 4*sqrt(a)*sqrt(2*sqrt(a)*sqrt(c + e) + b + d)*atan((sqrt(2*sqrt(a )*sqrt(c + e) - b - d) - 2*sqrt(c + e)*x)/sqrt(2*sqrt(a)*sqrt(c + e) + b + d))*c - 4*sqrt(a)*sqrt(2*sqrt(a)*sqrt(c + e) + b + d)*atan((sqrt(2*sqrt(a )*sqrt(c + e) - b - d) - 2*sqrt(c + e)*x)/sqrt(2*sqrt(a)*sqrt(c + e) + b + d))*e - 2*sqrt(c + e)*sqrt(2*sqrt(a)*sqrt(c + e) + b + d)*atan((sqrt(2*sq rt(a)*sqrt(c + e) - b - d) + 2*sqrt(c + e)*x)/sqrt(2*sqrt(a)*sqrt(c + e) + b + d))*b - 2*sqrt(c + e)*sqrt(2*sqrt(a)*sqrt(c + e) + b + d)*atan((sqrt( 2*sqrt(a)*sqrt(c + e) - b - d) + 2*sqrt(c + e)*x)/sqrt(2*sqrt(a)*sqrt(c + e) + b + d))*d + 4*sqrt(a)*sqrt(2*sqrt(a)*sqrt(c + e) + b + d)*atan((sqrt( 2*sqrt(a)*sqrt(c + e) - b - d) + 2*sqrt(c + e)*x)/sqrt(2*sqrt(a)*sqrt(c + e) + b + d))*c + 4*sqrt(a)*sqrt(2*sqrt(a)*sqrt(c + e) + b + d)*atan((sqrt( 2*sqrt(a)*sqrt(c + e) - b - d) + 2*sqrt(c + e)*x)/sqrt(2*sqrt(a)*sqrt(c + e) + b + d))*e + sqrt(c + e)*sqrt(2*sqrt(a)*sqrt(c + e) - b - d)*log( - sq rt(2*sqrt(a)*sqrt(c + e) - b - d)*x + sqrt(c + e)*x**2 + sqrt(a))*b + sqrt (c + e)*sqrt(2*sqrt(a)*sqrt(c + e) - b - d)*log( - sqrt(2*sqrt(a)*sqrt(c + e) - b - d)*x + sqrt(c + e)*x**2 + sqrt(a))*d - sqrt(c + e)*sqrt(2*sqr...